Directionally Regularized Two-Stage Calibration for Cross-Shaped MGT Sensor Arrays With Physics-Inspired Error Modeling

Accurate calibration of magnetic gradient tensor (MGT) sensor arrays requires correcting both the magnitude and directional responses of individual triaxial magnetometers. Conventional scalar-based calibration methods focus solely on fitting the magnitude of the magnetic field vector, lacking control over directional accuracy and thereby compromising the reliability of MGT construction under field conditions. To overcome these limitations, this paper proposes a directionally regularized two-stage calibration framework specifically for cross-shaped MGT fluxgate sensor arrays. A physically interpretable twelve-parameter error model is formulated to decouple sensor-internal distortions from array-level orientation misalignments. Crucially, a directional regularization term is incorporated into the calibration objective function to enforce vector alignment across the array. Furthermore, a high-precision triaxial fluxgate sensor is employed as an internal reference, enabling fully self-contained calibration without external field generators or scalar magnetometers. For accurate estimation of the calibration parameters, a two-stage optimization strategy based on Differential Evolution (DE) is developed, in which all parameters are coarsely estimated via magnitude consistency in the first stage, followed by refined alignment of array orientations via directional regularization in the second stage. Field experiments conducted over six independent trials validate the effectiveness and reproducibility of the proposed framework, yielding an overall mean magnitude RMSE of approximately 5.92 nT and a mean angular deviation of 0.07°. These results confirm that the proposed framework enables accurate and robust compensation of MGT sensor arrays in practical environments.